0,2

Amiram Eldar, Table of n, a(n) for n = 0..500

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

a(n) = n^2 * A005810(n).

a(n) = n * A378802(n).

a(n) == 0 (mod 4).

Sum_{n>=1} 1/a(n) = -(3/2)*log((c-1)/(c+1))^2 + (3/4) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1))^2 + (3/4) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1))^2 = 0.25947076781691783..., where c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013, p. 336, Example 3).

Sum_{n>=1} (-1)^n/a(n) = -(3/2)*log((1-d)/(1+d))^2 + (3/4) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1))^2 + (3/4) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1))^2 = -0.24154452788843591937..., where d = sqrt(1 - (8/sqrt(3))*(((3*sqrt(3)+sqrt(283))/16)^(1/3) - (((3*sqrt(3)+sqrt(283))/16)^(-1/3)))) (Batir and Sofo, 2013, pp. 336-337, Example 4).

a[n_] := n^2 * Binomial[4*n, n]; Array[a, 20, 0]

(PARI) a(n) = n^2 * binomial(4*n, n);

Cf. A005810, A378802.

Sequence in context: A181485 A135917 A241798 * A158450 A063406 A385977

Adjacent sequences: A378800 A378801 A378802 * A378804 A378805 A378806

nonn,easy

Amiram Eldar, Dec 07 2024

approved